Updated code

<?php
    $parent = get_cat_ID("photos");
    $cats = get_categories("child_of=".$parent);

    echo '<pre>';
        print_r( $cats );
    echo '</pre>';

    foreach ($cats as $cat) {
        query_posts( array (
            'showposts' => 1,
            'orderby' => 'rand',
            'cat' => $cat->cat_ID
    ) );
?>
<?php if ( have_posts() ) : while ( have_posts() ) : the_post(); ?>
    <ul>
        <?php if (has_post_thumbnail()) { ?>
        <?php
            printf( '<li><a href="'.get_category_link($cat->term_id).'">'.get_the_post_thumbnail(get_the_ID(), "album-thumb").'</a><a href="'.get_category_link($cat->term_id).'">%s</a></li>',
            apply_filters('get_term', $cat->name));
?>
        <?php } ?>
    </ul>
<?php endwhile; else: ?>
    <p><?php _e("Sorry, no posts matched your criteria."); ?></p>
<?php endif; } ?>
<?php wp_reset_query(); ?>